Compare Unusual Values in Both Groups
In Comparing Shapes of Two Distributions, you learned to compare features such as skewness and modality while naming both groups and the variable. This tutorial focuses on another feature: outliers. A useful comparison states whether either group has values that stand apart, identifies which group has them, and locates them in context.
An outlier is unusual relative to the distribution it belongs to. Therefore, assess the observations in each group using that group’s own scale of spread; do not decide that a value is an outlier merely because it is larger than values in the other group. As in What Outliers Mean in Context, a value flagged as an outlier deserves attention, but the flag alone does not tell you why it is unusual or whether it is an error.
A comparison might say, “For delivery times, the morning group has one high outlier, whereas the afternoon group has no values flagged as outliers.” This identifies the variable, both groups, and the difference. If a display marks the value, you can also report its approximate location and units.
Be precise about what “absent” means. If you checked the raw data against an outlier rule, you can say that no observations in that group fall beyond the rule’s fences. If you are interpreting a graph, it is often safer to say “no outliers are shown” or “no clear outliers are visible.” A graph may not show every individual observation, and an unmarked value is not automatically proof that no unusual value exists.
Use Each Group’s Own 1.5 IQR Fences
As introduced in What Outliers Mean in Context, the 1.5 IQR rule flags observations beyond the fences. The fences are calculated separately for each group using its first quartile, third quartile, and IQR. A value below the lower fence or above the upper fence is flagged. A value equal to a fence is not beyond it.
The fences are a consistent rule for identifying possible outliers, not boundaries that observations must stay inside. The lower fence can even be below zero when the measured variable cannot be negative; that does not mean the data are invalid. Compare each group’s observations with its own fences, then describe the results in context.
A boxplot often identifies outliers with individual points beyond the whiskers. In a standard boxplot, whiskers extend to the most extreme observations that are not flagged; a separate point represents an observation beyond a fence. Check the graph’s key or description before interpreting its symbols. If it does not show individual points or explain how outliers are marked, do not claim more than the display supports.
Confirm that both displays show the same variable, use the same units, and have a scale that lets you compare the groups.
Use its own boxplot marks or calculate its own IQR and fences. Do not use one group’s fences to judge the other group’s observations.
State whether the outlier is low or high and report an approximate value with units when the display supports that level of detail.
Name the variable and both groups, say which group has an outlier, and describe what is shown for the other group.
Worked Example: One High Outlier and None in the Other Group
Worked Example: One High Outlier and None in the Other Group
A fictional transit team records passenger waiting times, in minutes, at two service desks. Each group has ten observations. Use the 1.5 IQR rule to determine whether either group has outliers, then compare the results.
Desk A data: 12, 13, 14, 15, 16, 17, 18, 19, 20, 46.
Desk B data: 13, 14, 15, 16, 17, 18, 19, 20, 21, 22.
State. The question is whether passenger waiting times include any values flagged as outliers at either desk, and how the groups compare.
Plan. Both lists measure the same variable in minutes. For each group, find \(Q_1\), \(Q_3\), and the IQR, then calculate that group’s lower and upper fences. For ten ordered observations, use the median of the lower five observations for \(Q_1\), and the median of the upper five for \(Q_3\).
Do: Desk A. The lower five values are 12, 13, 14, 15, 16, so \(Q_1=14\). The upper five are 17, 18, 19, 20, 46, so \(Q_3=19\). Thus \(\text{IQR}=19-14=5\) minutes. The fences are \(14-1.5(5)=6.5\) and \(19+1.5(5)=26.5\) minutes. The value 46 is above 26.5, so Desk A has one high outlier, at 46 minutes. The other values are within the fences.
Do: Desk B. The lower five values are 13, 14, 15, 16, 17, giving \(Q_1=15\). The upper five are 18, 19, 20, 21, 22, giving \(Q_3=20\). Thus \(\text{IQR}=20-15=5\) minutes. Its fences are \(15-1.5(5)=7.5\) and \(20+1.5(5)=27.5\) minutes. Every Desk B observation is between 7.5 and 27.5 minutes, so none is flagged by this rule.
Conclude in context. “For passenger waiting times, Desk A has one high outlier at 46 minutes, whereas Desk B has no observations flagged as outliers by the 1.5 IQR rule.”
The conclusion names the variable and both desks, identifies the high value and its units, and explains what “none” means: no Desk B value is beyond its own fences. It does not claim that the unusual wait was an error.
An Outlier Is Relative to Its Own Group
An observation can be more extreme numerically in one group and still not be flagged as an outlier there. That can happen when the groups have different centers or spreads, because their quartiles and fences differ. Conversely, a value that is smaller than another group’s largest value can be an outlier if it is far from the rest of its own group.
Worked Example: The Larger Value Is Not Always the Outlier
A fictional repair center compares the number of hours needed to finish repair tickets in two work teams. Boxplots and summaries show that Team A has \(Q_1=10\) hours and \(Q_3=20\) hours, with a largest ordinary value of 30 hours and a separate point at 40 hours. Team B has \(Q_1=20\) hours and \(Q_3=35\) hours, and its greatest observed value is 42 hours. Use the 1.5 IQR rule to compare outliers.
Team A. Its IQR is \(20-10=10\) hours. The upper fence is \(20+1.5(10)=35\) hours, and the lower fence is \(10-1.5(10)=-5\) hours. The plotted point at 40 hours is above 35, so it is a high outlier. The largest ordinary value, 30 hours, is within the fences.
Team B. Its IQR is \(35-20=15\) hours. The upper fence is \(35+1.5(15)=57.5\) hours, and the lower fence is \(20-1.5(15)=-2.5\) hours. The greatest observed value, 42 hours, is below the upper fence; the values shown are not beyond either fence. Team B has no outliers flagged by this rule.
Compare in context. “For repair-ticket completion times, Team A has one high outlier at 40 hours, whereas Team B has no values flagged by the 1.5 IQR rule, even though its maximum of 42 hours is larger.”
This comparison avoids deciding that 42 must be the outlier simply because it is the largest number. Team B’s values are more spread out across the middle half, so its upper fence is higher. Whether an observation is flagged depends on its position relative to its own group’s distribution.
When a Display Shows an Outlier in Only One Group
Sometimes a question gives side-by-side boxplots rather than raw data. If the boxplot’s key identifies separate points as outliers, describe the points shown and compare the groups. You may not be able to calculate exact fences without quartiles, or identify exact values if the graph’s scale is coarse. In that case, report an approximate location only when it can be read from the scale.
A boxplot’s whisker endpoints also matter. The whisker does not necessarily end at the minimum or maximum observation; it typically reaches the most extreme non-outlier observation. So a separate point beyond a whisker is evidence of a flagged observation in that group, not evidence that the other group has the same value hidden at its whisker.
If one group has no separate points, match your wording to what you have actually examined. “The boxplot shows one outlier for Group A and none for Group B” is a direct description of the display. “The 1.5 IQR rule flags an observation in Group A but none in Group B” is appropriate when the rule has been applied to the data or the graph clearly uses that rule. Avoid the broader claim “Group B has no unusual values” unless you have defined and checked what counts as unusual.
Worked Example: Describe What a Boxplot Shows
A fictional parks department compares the time, in minutes, visitors spend at two community gardens. Side-by-side boxplots use the same scale. Garden North has one separate point near 95 minutes above its upper whisker. Garden South has no separate points, and its upper whisker reaches about 80 minutes. The graph does not provide the raw observations. Write a comparison supported by the display.
Read the evidence. The separate point indicates a high outlier shown for Garden North. The absence of separate points for Garden South means the boxplot shows no outliers there. The scale allows an approximate location for North’s point, but the display alone does not give enough information to recalculate either group’s fences.
Write the comparison. “For visitor time at the gardens, the boxplot shows one high outlier for Garden North at about 95 minutes, whereas it shows no outliers for Garden South.”
The phrase “the boxplot shows” is important because the raw data and the exact calculation are not provided. The sentence does not assert that North’s visitor stayed longer than every South visitor unless the graph supports that claim, and it does not suggest that an outlier must be a recording mistake.
Common Mistakes and AP Exam Tips
- Using one group’s fence for both groups. Calculate the IQR and fences separately. A group with greater spread can have wider fences, so its outlier assessment may differ.
- Calling the largest observation an outlier automatically. The maximum is simply the largest value. It is flagged by the 1.5 IQR rule only if it is above that group’s upper fence.
- Claiming “no unusual values” from a graph with no marked points. Describe what the display shows, or state that no observations are flagged by the rule you checked. Do not imply more certainty than the evidence provides.
- Calling an outlier an error. A flag means the observation is unusual relative to the group under a stated rule. It may be accurate; its source and context should be investigated before deciding what to do with it.
- Leaving out the variable or group names. “There is one outlier” is incomplete in a comparison. Identify the variable, name the group with the outlier, and describe the finding for the other group.
- Reporting a value more precisely than the graph allows. If a point is only readable as approximately 95 minutes, do not report it as exactly 95.0 minutes.
A full-credit comparison is specific and supported. For example, state that one group has a high or low outlier, identify its approximate value and units if available, and say that the other group has no outliers shown or no values beyond its own fences. The word “whereas” can make the contrast clear, but the evidence and context matter more than the particular transition word.
Check Your Understanding
Answer using the display information or apply the 1.5 IQR rule as specified.
- Group A has \(Q_1=8\) and \(Q_3=16\). Find its IQR and upper fence.
- For Question 1, Group A’s greatest value is 30. Is it flagged as a high outlier by the 1.5 IQR rule? Show the comparison.
- Group A has a high outlier at 50 minutes, while Group B has no values beyond its own fences. Write a contextual comparison about package-processing times.
- A side-by-side boxplot shows a separate point for Group East and no separate points for Group West. Why might “the boxplot shows no outliers for Group West” be more careful than “Group West has no unusual values”?
- Team A’s maximum is 40 hours and is flagged as an outlier. Team B’s maximum is 45 hours and is not flagged. Explain why these statements can both be true.